Ch. 4 - Probability

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 4 - Probability

Problem 4.4.25a

Chapter 4, Problem 4.4.25a

Is the Researcher Cheating? You become suspicious when a genetics researcher “randomly” selects numerous groups of 20 newborn babies and seems to consistently get 10 girls and 10 boys. The researcher claims that it is common to get 10 girls and 10 boys in such cases.

a. If 20 newborn babies are randomly selected, how many different gender sequences are possible?

Verified step by step guidance

Step 1: Understand the problem. The researcher is selecting 20 newborn babies and we are tasked with determining the number of different gender sequences possible. Each baby can either be a boy (B) or a girl (G), making this a problem of binary outcomes.

Step 2: Recognize that the number of different gender sequences corresponds to the total number of combinations of boys and girls in the group. Since each baby has two possible outcomes (boy or girl), the total number of sequences can be calculated using the formula for permutations with repetition: $ 2^{n} $, where $ n $ is the number of babies.

Step 3: Substitute $ n = 20 $ into the formula $ 2^{n} $. This gives $ 2^{20} $, which represents the total number of possible gender sequences for 20 babies.

Step 4: To interpret this result, note that $ 2^{20} $ accounts for every possible arrangement of boys and girls, including sequences like all boys, all girls, and any mix in between.

Step 5: Conclude that the calculation of $ 2^{20} $ provides the total number of gender sequences possible, which is a very large number, reflecting the vast number of combinations that can occur when selecting 20 newborns.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Random Selection

Random selection is a fundamental concept in statistics that refers to the process of choosing individuals from a population in such a way that each individual has an equal chance of being selected. This ensures that the sample is representative of the population, allowing for valid inferences to be made. In the context of the question, it implies that the selection of newborns should not favor one gender over the other.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and permutations of objects. In this scenario, it helps determine the number of different sequences of genders that can occur when selecting 20 newborns. Specifically, the number of gender sequences can be calculated using the formula for combinations, which accounts for the different ways boys and girls can be arranged in a sequence.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this case, it relates to the chance of obtaining a specific gender distribution (e.g., 10 boys and 10 girls) when randomly selecting 20 newborns. Understanding probability is crucial for evaluating whether the researcher's results are statistically significant or if they could occur by chance.

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